Information capacity of time-continuous channels

Abstract

The maximum average mutual information in the observation of the output, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y(t)</tex> , of a channel over the time interval <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[T_3,T_4]</tex> about the signal (input), <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s(t)</tex> , in the interval <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[T_1, T_2]</tex> is taken as the definition of channel capacity for the time-continuous case. In the case where the channel introduces additive Independent Gaussian noise of known correlation function, the capacity is evaluated subject to the constraint that the signal process have a given correlation function. For this evaluation a new joint expansion of the processes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s(t)</tex> is introduced which has the property that all coefficients in the expansion are uncorrelated. Thus, the expansion is a generalization of the Karkunen-Lo'{e}ve expansion to which it reduces when the noise is white and the time intervals coincide. The channel capacity is shown to be directly related to results in the theory of optimum filtering over a finite time interval. Closed form results for the capacity of several channels are given as well as some limiting expressions and bounds. For the case of white noise of spectral density <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N_o</tex> , the capacity is always bounded by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\bar{E}/N_o</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\bar{E}</tex> is the average signal energy.